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Convolution of periodic multiplicative functions and the divisor problem
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 14 November 2024, pp. 1-19
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We study a certain class of arithmetic functions that appeared in Klurman’s classification of
$\pm 1$ multiplicative functions with bounded partial sums; c.f., Comp. Math. 153(2017), 2017, no. 8, 1622–1657. These functions are periodic and 
$1$-pretentious. We prove that if 
$f_1$ and 
$f_2$ belong to this class, then 
$\sum _{n\leq x}(f_1\ast f_2)(n)=\Omega (x^{1/4})$. This confirms a conjecture by the first author. As a byproduct of our proof, we studied the correlation between 
$\Delta (x)$ and 
$\Delta (\theta x)$, where 
$\theta $ is a fixed real number. We prove that there is a nontrivial correlation when 
$\theta $ is rational, and a decorrelation when 
$\theta $ is irrational. Moreover, if 
$\theta $ has a finite irrationality measure, then we can make it quantitative this decorrelation in terms of this measure.
